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The twisted cubic in PG(3,q) and translation spreads in H(q)

G. Bonoli, O. Polverino
2005 Discrete Mathematics  
Using the connection between translation spreads of the classical generalized hexagon H (q) and the twisted cubic of PG(3, q), established in [European J.  ...  Combin. 23 (2002) 367-376], we prove that if q n ≡ 1 (mod 3), q odd, q 4n 2 − 8n + 2 and n > 2, then H (q n ) does not admit an F q -translation spread.  ...  3, q n ) of rank 2n whose points belong to imaginary chords of a twisted cubic C of PG(3, q n ), and conversely.  ... 
doi:10.1016/j.disc.2005.03.010 fatcat:3a4xumcd6bfctdv4sol43bq2ci

On the Twisted Cubic of PG(3, q)

G. Lunardon, O. Polverino
2003 Journal of Algebraic Combinatorics  
In this paper we classify the lines of PG(3, q) whose points belong to imaginary chords of the twisted cubic of PG(3, q).  ...  Relying on this classification result, we obtain a complete classification of semiclassical spreads of the generalized hexagon H (q).  ...  Let¯ be the twisted cubic of PG(3, F) defined by , where F is the algebraic closure of GF(q).  ... 
doi:10.1023/b:jaco.0000011940.77655.b4 fatcat:kkjzhp4yl5ecfhi5f357o73bjq

An investigation of the tangent splash of a subplane of $$\mathrm{PG}(2,q^3)$$ PG ( 2 , q 3 )

S. G. Barwick, Wen-Ai Jackson
2014 Designs, Codes and Cryptography  
2, q 3 ) in PG(6, q).  ...  In PG(2, q 3 ), let π be a subplane of order q that is tangent to ∞ . The tangent splash of π is defined to be the set of q 2 + 1 points on ∞ that lie on a line of π.  ...  One for pointing out the important relationship between tangent splashes and linear sets, and that a number of our results could be proved more directly using the theory of linear sets.  ... 
doi:10.1007/s10623-014-9971-3 fatcat:4yosskmsavhrxl3rvotiucbhgq

Absolute points of correlations of $$PG(3,q^n)$$

Giorgio Donati, Nicola Durante
2020 Journal of Algebraic Combinatorics  
As an application we show that, for q even, some of these sets are related to the Segre's (2 h + 1)-arc of PG(3, 2 n ) and to the Lüneburg spread of PG(3, 2 2h+1 ).  ...  In this paper, we completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space PG(3, q n ).  ...  Degenerate -quadrics and Lüneburg spread of PG(3, 2 n ). In 1965, H.  ... 
doi:10.1007/s10801-020-00970-3 fatcat:gtz2spm5r5cz3o2e566prrimou

An investigation of the tangent splash of a subplane of PG(2,q^3) [article]

S.G. Barwick, Wen-Ai Jackson
2014 arXiv   pre-print
2,q^3) in PG(6,q).  ...  In PG(2,q^3), let π be a subplane of order q that is tangent to ℓ_infty. The tangent splash of π is defined to be the set of q^2+1 points on ℓ_infty that lie on a line of π.  ...  One for pointing out the important relationship between tangent splashes and linear sets, and that a number of our results could be proved more directly using the theory of linear sets.  ... 
arXiv:1303.5509v2 fatcat:mluj5kn6e5he3b2524356ob444

Ruled cubic surfaces in PG(4,q), Baer subplanes of PG(2,q2) and Hermitian curves

Rey Casse, Catherine T. Quinn
2002 Discrete Mathematics  
The Andrà e=Bruck and Bose representation of PG(2; q 2 ) involves a regular spread in PG(3; q).  ...  (Utilitas Math. 17 (1980) 65) that non-a ne Baer subplanes of PG(2; q 2 ) are represented by certain ruled cubic surfaces in the Andrà e=Bruck and Bose representation of PG(2; q 2 ) in PG(4; q) (Math.  ...  Hence H ∈ PGL(2; q), so that the ruled cubic surface so determined is a V 3 2 contained in PG(4; q).  ... 
doi:10.1016/s0012-365x(01)00182-0 fatcat:27qjavb25rd5zaoleb77fbz4ra

Fq-pseudoreguli of PG(3,q3) and scattered semifields of order q6

Michel Lavrauw, Giuseppe Marino, Olga Polverino, Rocco Trombetti
2011 Finite Fields and Their Applications  
The known examples of such semifields are some Knuth semifields, some Generalized Twisted Fields and the semifields recently constructed in Marino et al. (in press) [12] for q ≡ 1 (mod 3).  ...  Here, we construct new infinite families of rank two scattered semifields for any q odd prime power, with q ≡ 1 (mod 3); for any q = 2 2h , such that h ≡ 1 (mod 3) and for any q = 3 h with h ≡ 0 (mod 3  ...  A way to construct scattered F q -linear sets of rank 6 in PG(3, q 3 ) is the following. Let r and r be two disjoint lines of PG(3, q 3 ) = PG(V ), say r = PG(U ) and r = PG(U ), with V = U ⊕ U .  ... 
doi:10.1016/j.ffa.2010.12.001 fatcat:etoffp2by5d53eygagj3cbcmlm

The tangent splash in (6,q) [article]

S.G. Barwick, Wen-Ai Jackson
2013 arXiv   pre-print
In the Bruck-Bose representation of PG(2,q^3) in PG(6,q), we investigate the interaction between the ruled surface corresponding to B and the planes corresponding to the tangent splash of B.  ...  Let B be a subplane of PG(2,q^3) of order q that is tangent to ℓ_∞. Then the tangent splash of B is defined to be the set of q^2+1 points of ℓ_∞ that lie on a line of B.  ...  If q is even, then the tangents to a twisted cubic lie in a regulus [9, q 3 ) , H * contains the twisted cubic N * , and so H * contains the transversal points P, P q , P q 2 .  ... 
arXiv:1305.6674v1 fatcat:tsrlqyus2rgpbb2wtd3h6se6n4

The exterior splash in PG(6,q): Special conics [article]

S.G. Barwick, Wen-Ai Jackson
2014 arXiv   pre-print
Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry CG(3,q), and hyper-reguli of PG(5,q).  ...  In this article we use the Bruck-Bose representation in PG(6,q) to give a geometric characterisation of special conics of π in terms of the covers of the exterior splash of π.  ...  In PG(6, q), [C i ] is an X-special twisted cubic in a 3-space Π i about a plane of X. By Theorem 3.10, the line P i L ii is in the 3-space Π i , hence Π i = Σ ii .  ... 
arXiv:1410.4269v1 fatcat:exrug5lff5hptna72c4aeaww6e

The exterior splash in PG(6,q): Transversals [article]

S.G. Barwick, Wen-Ai Jackson
2014 arXiv   pre-print
In PG(6,q), an exterior splash S has two sets of cover planes (which are hyper-reguli) and we show that each set has three unique transversals lines in the cubic extension PG(6,q^3).  ...  Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry CG(3,q), and hyper-reguli in PG(5,q).  ...  An S-special twisted cubic is a twisted cubic N in a 3-space of PG(6, q)\Σ ∞ about a plane of S, such that the extension of N to PG(6, q 3 ) meets the transversals of S.  ... 
arXiv:1409.6794v1 fatcat:j5kz5cbsynadxo3iwzuiqz4dye

On symplectic semifield spreads of PG(5,q 2), q odd

Giuseppe Marino, Valentina Pepe
2018 Forum mathematicum  
We prove that there exist exactly three non-equivalent symplectic semifield spreads of {\operatorname{PG}(5,q^{2})} , for {q^{2}>2\cdot 3^{8}} odd, whose associated semifield has center containing {\mathbb  ...  Equivalently, we classify, up to isotopy, commutative semifields of order {q^{6}} , for {q^{2}>2\cdot 3^{8}} odd, with middle nucleus containing {\mathbb{F}_{q^{2}}} and center containing {\mathbb{F}_{  ...  associated with a twisted field A and a spread comprising the constructions BH, LMPT , ZP.  ... 
doi:10.1515/forum-2016-0133 fatcat:2fg7js2465eptofbpkkcec7qta

On projective q^r-divisible codes [article]

Daniel Heinlein, Thomas Honold, Michael Kiermaier, Sascha Kurz and Alfred Wassermann
2019 arXiv   pre-print
One example are upper bounds on the cardinality of partial spreads. Here we survey the known results on the possible lengths of projective q^r-divisible linear codes.  ...  Especially, q^r-divisible projective linear codes, where r is some integer, arise in many applications of collections of subspaces in F_q^v.  ...  x) = f , and (3) min{ x∈H w(x) | HH} = m, where H is the set of hyperplanes of PG(v − 1, q) .  ... 
arXiv:1912.10147v1 fatcat:fpig43peijgvrn5dngi5aaa6q4

Page 2702 of Mathematical Reviews Vol. , Issue 2003d [page]

2003 Mathematical Reviews  
By representing H(q) as a coset geometry, we obtain a characterization of a translation spread in terms of a set of points of PG(3,q) which belong to imaginary chords of a twisted cubic and we construct  ...  Let 2, = PG(3,q) be the hyperplane at infinity, and let L be the fixed regular spread of x. used in the above representation.  ... 

Page 10043 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
(I-NAPL2; Caserta) On the twisted cubic of PG(3,q¢). (English summary) J. Algebraic Combin. 18 (2003), no. 3, 255-262.  ...  Lines of PG(3,q) whose points belong to imaginary chords of the twisted cubic £ of PG(3,q) are classified by proving that if / is a line whose points belong to imaginary chords of £, then either / is an  ... 

Linear sets in finite projective spaces

Olga Polverino
2010 Discrete Mathematics  
Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids  ...  In this paper linear sets of finite projective spaces are studied and the "dual" of a linear set is introduced.  ...  Acknowledgement This work was supported by the Research Project of MIUR (Italian Office for University and Research) ''Strutture geometriche, combinatoria e loro applicazioni''.  ... 
doi:10.1016/j.disc.2009.04.007 fatcat:tuvgbx5qtbgkxmn74ywas2aysm
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